Dynamic Behavior Analysis Of A Class Of Infinite Dimensional Coupling System With Damping

With the wide application of viscoelastic materials on mechanics, chemical engi-neering, architecture, trafc, information and so on, the dynamic behavior and controlof vibration for elastic structures with viscoelasticity have attracted much attention ofacademic and engineer circles over the past several decades. Since temperature is one ofthe most important factors of infuencing the viscoelastic property of materials, hencethe mathematical models we obtained are usually infnite-dimensional hybrid systems,which couple the heat conduction equation with the heat elastic equation. Vibration re-duction technology based on the viscoelastic damping materials has made considerableprogress since the1960s, it has been extensively applied in vibration and noise control ofvarious military profession, space fights, aviation, warships, etc.. Therefore, the studyon control and stabilization of infnite-dimentional coupled systems with dampings isinstructive.By means of operator semigroup theory and the technique of asymptotic analysis,this thesis studies the dynamic behavior of infnite-dimentional coupled systems withdampings by using the method of spectral analysis and Riesz basis approach, which aretypical distributed parameter systems described by functional diferential equations orPDEs. This thesis is divided into three parts: the dynamic behavior of a single waveequation with viscoelastic and viscous dampings are studied in the frst part, whichis the content of Chapter2. The Riesz basis property and exponential stability ofa class of PDE PDE infnite-dimentional coupled systems are studied in the secondpart, which includes the content of Chapter3-Chapter5. In the last part, i.e. Chap-ter6, the boundary feedback control and exponential stabilization of a PDE ODEinfnite-dimentional coupled systems are discussed. Specifcally, this thesis is orginizedas follows:In Chapter1, a review of theories of viscoelasticity, thermoviscoelasticity and ther- moelasticity, which occupy important positions in material science, are presented. The structure and main results of this thesis are briefly summarized, and we also give some concepts and theoretical results to be used in the thesis.In Chapter2, we study the spectral analysis and exponential stability of a sin-gle one-dimensional wave equation with Boltzmann viscoelastic and viscous friction dampings Firstly, some new variables are introduced to transform the system into a time-invariant one, then by defining an unbounded operator, we transform the newly formulated sys-tem into an abstract evolution equation in the Hilbert state space, and prove the well-posedness of the system by using the knowledge of functional analysis. Secondly, with the technique of asymptotic analysis, we give the asymptotic expressions for vibration frequencies. Finally, the Riesz basis generation is verified and consequently the expo-nential stability of the system is deduced, this shows that the dynamics of the vibrating system is determined completely by the vibration frequencies. The result shows that the viscoelastic damping is strong enough to produce an exponential stability for the system.In Chapter3, we study a one-dimensional thermviscoelastic system with Dirichlet-Dirichlet boundary conditions: This system describes the motion of a viscoelastic rod (or bar) affected by the tem-perature, there exist the interaction and influence between the deformation and the temperature of the rod, hence the heat conduction equation and the heat elastic e-quation is coupled together rather than independent on each other. It is equivalent to the following hyperbolic-parabolic infinite-dimentional coupled system with viscoelastic damping: By using semigroup theory and spectral analysis method, we show the well-posedness of the system, discuss the asymptotic distribution of the spectrum for the system operator and verify the Riesz basis property. As a result, the spectrum-determined growth condition holds true and then the system is stable in an exponential manner when the parameters k,μ satisfying k≠μ. It shows that:the dissipative mechanism of heat conduction and viscoelastic damping is strong enough to make the semigroup associated with the system not only exponential stable but also analytic, that is to say, the whole system with thermoviscoelastic damping can be considered as a dynamical compensator to stabilize itself with parameters k≠μ.In Chapter4, based on Chapter3, we replace the higher order term with a specific Boltzmann damping in the heat conduction equation to get the following PDE-PDE infinite-dimentional coupled systems by introducing some new variables: According to the semigroup theory and the spectral analysis, we analyze the well-posedness and spectrum distribution on the complex plane, and prove the Riesz basis property of the system operator, and thus we obtain the exponential stability of the system. Similarly, the whole system with thermoviscoelastic damping can be considered as a dynamical compensator to stabilize itself exponentially. It is interesting to note that the change in the heat conduction equation weaken the dissipation of the whole system, the property of the semigroup associated with the system becomes weaker accordingly, it is not analytic.In Chapter5, we address the dynamic behavior of an infinite-dimentional coupled system for thermoelasticity of type Ⅲ Here the heat conduction equation is hyperbolic instead of the parabolic equation de-rived by the classical theory of thermoelasticity. That is to say, the theory of thermoe-lasticity of type Ⅲ is developed in a rational way in order to produce a fully consistent explanation which is capable of incorporating thermal pulse transmission in a very log-ical manner and elevate the unphysical infinite speed of heat propagation induced by the classical theory of heat conduction. Mathematically, it is represented through a system of wave equation with K-V damping In this chapter, applying the technique of asymptotic analysis and spectral analysis method, we give the asymptotic expressions of the eigenvalues and eigenfunctions and verify the Riesz basis property of the system operator, accordingly, the exponential stability of the system is obtained. The results and the numerical simulations show that the dissipation only produced by the heat conduction equation is strong enough to stabilize the system exponentially, i.e., we can regard the wave equation with K-V damping as a dynamical compensator to stabilize the whole system.In Chapter6, by means of the Riesz basis method, we study the feedback con-trol and stabilization of the following Euler-Bernoulli Beam-ODE infinite-dimentional coupled system where, the forth-order beam equation can be considered as a controller, the original control plant ODE is connected with PDE through boundary output of the beam equa-tion. Firstly, we obtain the asymptotical expressions for eigenvalues and eigenfunctions by applying the technique of spectrum analysis. Secondly, we prove there is a set of gen-eralized eigenfunctions which forms a Riesz basis for the Hilbert state space, and then the spectrum-determined growth condition is established, consequently, the exponential stability of the system is obtained.